Question: Rania is riding the ferris wheel. Her vertical height $H(t)$ (in $\text{m}$ ) off the ground as a function of time $t$ (in seconds) can be modeled by a sinusoidal expression of the form $a\cdot\cos(b\cdot t)+d$. At $t=0$, when she starts moving, she is at a height of $10\text{ m}$ off the ground, which is as low as she goes. After $20\pi$ seconds, she reaches her maximum height of $30\text{ m}$. Find $H(t)$. $\textit{t}$ should be in radians. $H(t) = $
Solution: The strategy First, we should convert the given information about the real-world context into mathematical terms of the sinusoidal function and its graph. Then, we should use the given information to find the amplitude, midline, and period of the function's graph. Finally, we should find $a$, $b$, and $d$ in the expression $a\cos(b\cdot t)+d$ by considering the features we found. Converting the given information into mathematical terms At $t=0$, Rania is $10\text{ m}$ from the ground. This means the graph of the function passes through $(0,10)$. We are given that this is the closest point to the ground, which corresponds to a minimum point of the graph. $20\pi$ seconds later (which means $t=20\pi$ ) the her distance is $30\text{ m}$. This corresponds to the point $(20\pi,30)$. We are given that this is her maximum height off the ground, which corresponds to a maximum point of the graph. In conclusion, the graph has a minimum point at $(0,10)$ and then has a maximum point at $(20\pi,30)$. Determining the amplitude, midline, and period The midline intersection is halfway between the maximum and minimum, which is at $y={20}$, so this is the midline. The minimum point is $10$ units below the midline, so the amplitude is ${10}$. The maximum point is $20\pi$ units to the right of the nearest minimum, so the period is $2\cdot 20\pi={40\pi}$. [Why did we multiply by 2?] Determining the parameters in $a\cos(b\cdot t)+d$ Since the minimum at $t=0$ is followed by a maximum point, we know that $a<0$. [How do we know that?] The amplitude is ${10}$, so $|a|={10}$. Since $a<0$, we can conclude that $a=-10$. The midline is $y={20}$, so $d=20$. The period is ${40\pi}$, so $b=\dfrac{2\pi}{{40\pi}}=\dfrac{1}{20}$. The answer $H(t)=-10\cos\left(\dfrac{1}{20}t\right)+20$